MR 13 – Expectations, Risk Premium, Convexity, and the Shape of the Term Structure

Suppose the one-year rate today is 8%, and every investor knows, with complete certainty, that the one-year rate will be 7% a year from now and 6% two years from now. With annual periods and a unit face value, each zero coupon bond is simply its face value discounted along that known path of one-year rates.

A forward rate is defined as the rate that links the price of one zero to the price of the next longer zero, so P(1)/P(2) minus 1 is the one-year rate one year forward. Applying that definition to the three prices above returns exactly 8%, 7% and 6%. So, when there is no volatility around expectations, the forward curve is nothing more than the expected path of the short-term rate written out year by year. Because expectations could in principle follow any path, the term structure could be flat, rising, falling, or humped.

Why expectations cannot take any shape in practice

Market participants can hold quite specific views about short rates over the next few months. Central bank meeting dates, anticipated policy moves, tax payment dates, the calendar of bond issuance and quarter-end balance sheet management all shape the supply of and demand for funds over short horizons. Far out along the curve, however, that granularity disappears. No study of money market conditions will tell an analyst that the one-year rate expected in 29 years differs much from the one expected in 30 years.

At those long horizons, expectations tend to be anchored by macroeconomic reasoning instead. An analyst might argue, for example, that the short-term rate should settle at 4% in the long run, made up of a 1% long-run real rate of interest plus 3% of long-run inflation. So, expectations explain a great deal of the short end of the curve, while the long end flattens toward a long-run anchor.

Investors know their forecasts are imperfect, so realized short rates are modelled as random moves around expectations. Consider a binomial tree for the one-year rate with annual steps and a 50% probability on every branch. The rate starts at 9%, moves to 13% or 5% after one year, and then to 17%, 9% or 1% after two years. These levels and moves are deliberately exaggerated so that the effects are easy to see.

The expected rate is the same at every date. At date 1 it is 50% × 13% + 50% × 5% = 9%, and at date 2 it is 50% × [50% × 17% + 50% × 9%] + 50% × [50% × 9% + 50% × 1%] = 9%. The volatility of each transition is also constant at 4%, or 400 basis points.

Assume for now that investors are risk neutral, so each date-0 price is the expected discounted value of the next-date prices. The one-year zero is worth 1/1.09, or 0.917431. Working backward through the price trees gives the two-year zero at 0.842815 (date-1 values 0.884956 and 0.952381) and the three-year zero at 0.776366 (date-1 values 0.784129 and 0.908348; date-2 values 0.854701, 0.917431 and 0.990099).

Although expected rates are flat at 9%, the forward curve slopes downward, and the gap widens with maturity. Nothing about expectations creates that slope. It comes from volatility interacting with the curvature of bond prices, which the next section explains.

The price of a zero is a convex function of its rate. Jensen’s inequality says that, for a convex function of a random variable, the average of the function exceeds the function evaluated at the average. For bonds, it means that the expected price is greater than the price computed at the expected rate.

The tree makes the inequality concrete. A one-year zero priced at date 1 is worth either 0.884956 or 0.952381, with an average of 0.918668, while the price at the expected rate of 9% is only 0.917431. Now let f be the one-year rate one year forward. The two-year zero is the unit face discounted at 9% in the first year and at f in the second, and along the tree it is also the discounted expected date-1 price.

Multiplying through by 1.09 and invoking Jensen’s inequality gives the comparison below, and from it the one-year rate one year forward must be less than the 9% rate expected in one year.

So, volatility makes bonds more valuable than they would be at expected rates, and a higher price is the same thing as a lower forward rate. The size of the convexity effect can be estimated directly as the forward rate minus the expected rate: about 14.68 basis points at two years and 44.10 basis points at three years in this tree.

A continuous-time version of the same ideas starts by assuming that all bond prices depend on a single instantaneous rate r, whose value at time t is r_t, and writing P_t(r_t, T) for the price at time t of a zero maturing in T years. Ito’s lemma, which is used here without proof, gives the instantaneous price change.

The first partial derivative measures the response of price to the rate holding time fixed, the second measures the response to time holding the rate fixed, and the second-order term captures how the rate sensitivity itself changes with the rate. Dividing by P turns price changes into returns.

Rewriting the terms with forward rates, duration and convexity

With instantaneous forward rates f(t), a T-year zero is priced by compounding along the forward curve. Moving t forward shortens the remaining term T one for one, so differentiating gives the time derivative.

Duration and convexity are defined from the rate derivatives.

Substituting all three gives the return decomposition, and taking expectations gives its expected form.

The first component is the return earned purely from the passage of time, and for a zero it equals the forward rate for its maturity, f(T). This links to the familiar result that a zero earns its forward rate when the term structure stays unchanged and rates do not move. The second component says a rise in rates cuts the return in proportion to duration. The third says that volatility, whether rates move up or down, adds to the return in proportion to convexity, which is why, at a fixed instant and for the same duration, a more convex portfolio gains more from any rate move. In the expected form, the duration term depends on the expected change in the rate rather than the realized one.

Risk-neutral investors require no compensation for bearing interest rate risk, so they price every bond to offer the short-term rate as its expected return.

Risk-averse investors want more expected return from bonds that carry more interest rate risk. Over the next instant that risk is captured by duration with respect to the single rate factor, so the extra return demanded is proportional to duration. The risk premium per unit of duration may vary with time and with the level of rates, but it does not depend on the features of any individual bond. Treating it as a constant λ gives the expected return condition.

The risk premium as a Sharpe ratio

The Sharpe ratio (SR) of a security is its expected return in excess of the short-term rate divided by the standard deviation of its return. A bond’s random return comes from duration times the rate change, so the standard deviation of its return is duration times the standard deviation of rates. Dividing the excess return from the risk-averse condition by that quantity leaves the risk premium per unit of rate volatility.

With a risk premium of 10 basis points per year and a standard deviation of rates of 100 basis points per year, every bond has a Sharpe ratio of 10%. The ratio is identical across bonds because both the excess return and the risk grow in proportion to duration.

Setting the expected return from the decomposition equal to the expected return that risk-averse investors demand, and solving for f(T), separates the forward rate into three effects.

Reading the three terms

The expectations term starts from the instantaneous rate and adds the expected rate change scaled by the duration of the zero matching the forward’s term. A higher current rate, faster expected increases, and a longer duration all push the forward rate up. The risk premium term raises the forward rate in proportion to that same duration, so more interest rate risk and a larger premium both mean a higher forward rate. The convexity term lowers the forward rate by one half of C times σ squared, which is the analytical form of the Jensen’s inequality effect seen in the tree.

Because the first two terms both multiply D, a drift of a given number of basis points in the short-term rate prices bonds exactly as an equal-sized premium per unit of duration would. Forward rates alone therefore cannot tell expectations apart from risk premium. Only the risk-neutral process, which combines the two, matters for pricing; splitting it is useful for economic views and macro trading but cannot be observed from a cross section of bond prices.

Pure expectations versus pure risk premium hypothesis

Leaving convexity aside, the pure expectations hypothesis sets λ to zero, so forward rates reflect E[dr/dt] alone and the natural no-change scenario is that short rates evolve as expected and forwards are realized. The pure risk premium hypothesis sets E[dr/dt] = 0, so forwards reflect the premium alone and the natural no-change scenario is an unchanged term structure. Reality may lie anywhere between the two.

Convexity cuts both ways. It lowers the forward rate, which reduces the time component of return, yet it adds the same amount directly through the convexity term. Under the risk-averse expected return condition the two offsets cancel, so convexity confers no net advantage in expected return.